Why Tree Structures Limit Math Knowledge Graphs and How Network Thinking Helps

I have been researching K‑12 math knowledge graphs, aiming to structure scattered knowledge points and their relationships.

Advantages and Limitations of Tree Thinking

Initially I organized the curriculum into a tree hierarchy with four main branches: Numbers & Algebra, Geometry, Probability & Statistics, and Integrated Applications. Each branch is further divided by grade and logical order.

Benefits of this tree structure include:

Clear hierarchy : students quickly know their current level and the next step.

Explicit dependencies : most concepts have a identifiable prerequisite (e.g., quadratic functions depend on linear functions).

Easy management : curricula and learning plans can be mapped onto the tree.

However, the tree model has inherent limitations:

Cross‑knowledge connections are invisible : solving a probability problem may require functions, sequences, or geometry, which the tree forces into separate branches.

Slow knowledge iteration : new topics that don’t fit existing branches require forced extensions, complicating management.

Problem solving is not purely linear : learners often jump between concepts rather than follow a single downward path.

Tree thinking is deductive, strong for analysis and decomposition but weak for synthesis and innovation.

Introducing Network Thinking

To address these shortcomings, I adopted a network (graph) perspective for the knowledge graph.

Mathematically, a graph represents knowledge points as nodes and their prerequisite relationships as edges, allowing cycles, cross‑links, and multi‑directional connections. An adjacency matrix can capture these relationships.

Example of a probability knowledge network:

“Random event” node connects to “Set operations” because event operations are essentially set operations.

“Classical probability model” points to both “Permutations & Combinations” and the “Principle of Equal Likelihood”.

“Conditional probability” depends on the “Multiplication rule” and links to the “Bayes theorem” and “Statistical decision”.

Benefits of the Network Structure

Richer lateral connections : different branches can freely interconnect.

Matches real problem‑solving paths : students can explore solutions along any route.

Facilitates knowledge transfer : new problem types, often combinations of existing concepts, are easier to recognize.

Complementary Fusion of Tree and Network

Practice shows that the optimal solution is not to replace the tree entirely but to combine both:

Macro use of the tree : the overall curriculum is hierarchical, aiding planning and navigation.

Micro use of the network : at the level of specific concepts or problem types, the graph expresses cross‑links and multi‑directional dependencies.

Metaphorically, the tree is the “skeleton” of knowledge, while the network is the “vascular system” that ensures information flow.

Mathematics Fusion Model

Let the entire set of knowledge points be K. Two relations are defined:

Hierarchical relation, forming a tree.

Associative relation, forming a directed graph.

The resulting knowledge graph is a composite structure: the tree presents the curriculum hierarchy, and the graph reveals cross‑disciplinary connections, allowing students to follow the tree first and then traverse the network for integration.

Extended Implications

Tree thinking provides ordered, linear guidance suitable for curricula and foundational training, while network thinking offers associative, innovative pathways ideal for problem solving and interdisciplinary integration. Over‑reliance on either can lead to linear dependency traps or chaotic, unmanageable structures. Dynamic switching and structural fusion offer a more effective approach.

In mathematics education and knowledge management, students need to climb the tree for direction and navigate the network for flexibility—both are essential for a truly usable and growing math knowledge system.

Recommended reading: How Mathematics Can Be Learned by Shu‑gong Tsuruoka, which aligns with the ideas of combining tree and network thinking.